We observe that due to the convexity of \(l(\beta , \textbf {x})\)  in \(\beta \)  and the 1-strong convexity of the \(L_2\) -regularization term in (REF ), \(J(\beta ,D)\)  is \(\lambda \) -strongly convex. Hence, we obtain by Shalev-Shwartz inequality [1]}
\(\left(\nabla _{\beta } \ J(\beta _1^*,D) - \nabla _{\beta } \ J(\beta _2^*,D)\right)^T \left( \beta _1^* - \beta _2^*\right) \ge \lambda \Vert  \beta _1^* - \beta _2^* \Vert _2^2.\) 
